1. Materials Considerations in Semiconductor Detectors
S W McKnight and
C A DiMarzio

2. Electrons in Solids: Schrodinger’s Equation
Kinetic energy
Potential energy
Total energy
= electron wave function
= probability of finding electron between x and x+dx at time between t and t+dt
Normalization: Integral of ΨΨ* over all space and time=1

3. Wavefunction and Physical Observables
Momentum:
Energy:
(Planck’s constant)

4. Time-independent Schrodinger’s Equation
Separation of variables:

5. Solutions to Schrodinger’s Equation
Free particle: V=0
Solution:
Wave traveling to left or right with:

6. Free Particle

7. Periodic Potentials a
V(x) x
where:
= “crystal momentum”

8. Bloch Theorem
Since this holds for any x+a, adding or subtracting any number of reciprocal lattice vectors (2π/a) from crystal momentum does not change wavefunction. Can describe all electron states by considering k to lie in interval (π/a > k > -π/a) (first Brillouin zone)
Physical Interpretation: electron can exchange momentum with lattice in quanta of (2π/a)

9. “Empty Lattice” V=0, but apply lattice periodicity, V(r+a)=V(r) k E E

10. “Empty Lattice”: Reduced Zone
V=0, translated to First Brillouin Zone k E E k

11. Kronig-Penny Potential
V(x) x
ψ(x)

12. Band Gaps
V≠0 lifts degeneracy at band crossings k E Eg E Eg k

13. Electron States in Band
Electron state “phase space” volume: ΔpxΔx=h
Number of electron states per unit length (per spin) with –kf<k<kf = 2 kf / (2π)

14. Electron States in Band
Number electron states/unit length in band = [π/a – (-π/a)]/(2π)
= 1/a k E Eg E Eg k
Δk=2π

15. Photon Momentum vs. Crystal Momentum
Photon momentum is small compared to electron crystal momentum

16. Optical Band Transitions
Momentum conservation implies optical transitions in band are nearly vertical k E Eg E Eg k

17. Effective Mass Approximationk
Near minimum:
m*=effective mass

18. “Hole” Approximation E k Hole effective mass =mh* <0
Vacancy k
Band energy = Filled band – electron vacancy
Hole effective mass =mh* <0

19. Semiconductor Band Structures

20. Semiconductor Band Structures

21. Direct and Indirect Gaps
Direct-gap semiconductors
Electrons and holes at same k
Ge, GaAs, CdTe
Strong coupling with light, Δk≈0
Indirect-gap semiconductors
Electrons at different k than holes Si
Weak coupling with light, Δk≠0
Need phonon to conserve momentum
Multistep process: photon + electron(E, k) →
electron (E+hν, k) + phonon →
electron(E+hν, k+Δk)